Econometrics - Instrumental Variables

IVEstimationWhy use an IV estimator? Suppose that X and are not uncorrelated. Then least squares isneither unbiased nor consistent.Recall the proof of consistency of least squares:b = β + (X'X/n)-1(X's/n)Plim b = β requires plim(X's/n) = 0. If this doesnot hold, the estimator is inconsistent

IV Estimation Why use an IV estimator? Suppose that X and  are not uncorrelated. Then least squares is neither unbiased nor consistent. Recall the proof of consistency of least squares: b =  + (X’X/n)-1 (X’/n). Plim b =  requires plim(X’/n) = 0. If this does not hold, the estimator is inconsistent

APopularMisconceptionA popular misconception. If only one variable in x is correlated with ,the other coefficients are consistently estimated. False.Suppose onlythe first variable is correlated with6160ThenUnderthe assumptions,plim(x'e/n):gl1F189210plim b-β = plim(x'X/n)=01eK1js times the first column of Q-1The problem is"smeared"overtheother coefficients

A Popular Misconception A popular misconception. If only one variable in X is correlated with , the other coefficients are consistently estimated. False. The problem is “smeared” over the other coefficients. 1 11 1 21 -1 1 1 1 Suppose only the first variable is correlated with 0 Under the assumptions, plim( /n) = . Then . . 0 plim = plim( /n) . . . times K q q q                               =                =  ε X'ε b - β X'X -1 the first column of Q

TheGeneralResultBy construction, the IV estimator is consistent.So, we have an estimator that is consistentwhen least squares is not

The General Result By construction, the IV estimator is consistent. So, we have an estimator that is consistent when least squares is not